Measurement of small wavelength difference in coherent light using faraday effect

ABSTRACT

An apparatus is provided for determining a target wavelength λ of a target photon beam. The apparatus includes a photon emitter, a pre-selection polarizer, a prism composed of a Faraday medium, a post-selection polarizer, a detector and an analyzer. The photon emitter projects a monochromatic light beam at the target wavelength λ substantially parallel to a magnetic field having strength B. The target wavelength is offset from established wavelength λ′ as λ=λ′+Δλ by wavelength difference of Δλ&lt;&lt;λ. The Faraday prism has Verdet value V. After passing through the pre-selection polarizer, the light beam passes through the prism and is incident to an interface surface at incidence angle θ 0  to the normal of the surface and exits into a secondary medium as first and second circularly polarized light beams separated by target separation angle δ and having average refraction angle θ. The secondary medium has an index of refraction of n 0 . After passing the post-selection polarizer, the detector measures target pointer rotation angle A w  based on the target separation angle δ. The analyzer determines the target wavelength λ by calculating offset pointer rotation angle ΔA w =A w −A′ w  from calibrated pointer rotation angle A′ w  based on established separation angle δ′ that corresponds to the established wavelength λ′, and by estimating the wavelength difference based on 
     
       
         
           
             
               
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     in which ε is an amplification factor. A method is provided incorporating operations described for the apparatus.

CROSS REFERENCE TO RELATED APPLICATION

The invention is a Continuation-in-Part, claims priority to andincorporates by reference in its entirety U.S. patent application Ser.No. 13/134,486 filed Jun. 6, 2011 titled “Magnetic Field Detection UsingFaraday Effect” and assigned Navy Case 99670.

STATEMENT OF GOVERNMENT INTEREST

The invention described was made in the performance of official dutiesby one or more employees of the Department of the Navy, and thus, theinvention herein may be manufactured, used or licensed by or for theGovernment of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefor.

BACKGROUND

The invention relates generally to detection of small wavelengthdifferences. In particular, this invention relates to a quantum enhancedmethod for determining small differences in coherent photon wavelengthsusing weak value amplification of a Faraday Effect.

Detecting the presence of two nearly coincident wavelengths can becomenecessary for various operations, such as for detection of small Dopplershifts. However, conventional devices and processes exhibit deficienciesin cost and/or portability.

SUMMARY

Conventional wavelength discrimination devices yield disadvantagesaddressed by various exemplary embodiments of the present invention.Various exemplary embodiments provide an apparatus for determining atarget wavelength λ of a target photon beam wavelength λ using a prismcomposed of a Faraday medium having Verdet value V.

The apparatus includes a photon emitter, a pre-selection polarizer, aprism composed of a Faraday medium, a post-selection polarizer, adetector and an analyzer. The photon emitter projects a monochromaticlight beam at the target wavelength λ substantially parallel to amagnetic field having strength B. The target wavelength is offset fromestablished wavelength λ′ as λ=λ′+Δλ by wavelength difference of Δλ<<λ.

The light beam passes through the pre-selection polarizer and the prism.The beam is incident to an interface surface at incidence angle θ₀ tothe normal of the surface and is refracted into a secondary medium asfirst and second circularly polarized light beams separated byseparation angle δ and having average refraction angle θ. The secondarymedium has an index of refraction of n₀.

The two circularly polarized light beams pass through the post-selectionpolarizer and reach the detector, which measures target pointer rotationangle A_(w) based on the separation angle δ. The analyzer determines thewavelength difference Δλ, first by calculating offset pointer rotationangle ΔA_(w)=A_(w)−A′_(w) from calibrated pointer rotation angle A′_(w)based on the separation angle θ′ that corresponds to the establishedwavelength λ′, and second by estimating the wavelength difference basedon

${{\Delta \; \lambda} \approx {- \frac{2{ɛ\pi}\; n_{0}\Delta \; A_{w}\cos \; \theta^{\prime}}{{VB}\; \sin \; \theta_{0}}}},$

in which ε is an amplification factor. Various exemplary embodimentsalso provide a method that incorporates operations described for theapparatus.

BRIEF DESCRIPTION OF THE DRAWINGS

These and various other features and aspects of various exemplaryembodiments will be readily understood with reference to the followingdetailed description taken in conjunction with the accompanyingdrawings, in which like or similar numbers are used throughout, and inwhich:

FIG. 1 is a diagram view of an optical diagram; and

FIG. 2 is a schematic view of a wavelength discriminator apparatus.

DETAILED DESCRIPTION

In the following detailed description of exemplary embodiments of theinvention, reference is made to the accompanying drawings that form apart hereof, and in which is shown by way of illustration specificexemplary embodiments in which the invention may be practiced. Theseembodiments are described in sufficient detail to enable those skilledin the art to practice the invention. Other embodiments may be utilized,and logical, mechanical, and other changes may be made without departingfrom the spirit or scope of the present invention. The followingdetailed description is, therefore, not to be taken in a limiting sense,and the scope of the present invention is defined only by the appendedclaims.

In accordance with a presently preferred embodiment of the presentinvention, the components, process steps, and/or data structures may beimplemented using various types of operating systems, computingplatforms, computer programs, and/or general purpose machines. Inaddition, those of ordinary skill in the art will readily recognize thatdevices of a less general purpose nature, such as hardwired devices, orthe like, may also be used without departing from the scope and spiritof the inventive concepts disclosed herewith. General purpose machinesinclude devices that execute instruction code. A hardwired device mayconstitute an application specific integrated circuit (ASIC) or afloating point gate array (FPGA) or other related component.

This disclosure provides an overview of how a combined application of arecently discovered Faraday effect and weak value amplification can beused to measure Δλ=λ−λ′ (thereby detecting λ) when λ′ is known. Themethodology has potential spectroscopic utility in such areas asmeasuring small Doppler shifts and detecting the presence of otherwiseindistinguishable chemical or biological spectroscopic markers. Suchmeasurements can be made using either classically intense laser light orsingle photon streams. For example, let λ and λ′ be two wavelengths suchthat λ=λ′+Δλ with Δλ being small.

A longitudinal magnetic field induces a circular differential refractionof a linearly polarized photon beam at the boundary between a Faradaymedium and a medium with negligible Verdet constant as reported by A.Ghosh et al., “Observation of the Faraday effect . . . ” in Phys. Rev. A76, 055402 (2007). See eitherhttp://www.rowland.harvard.edukjf/fischedimages/PRA_(—)76_(—)055402.pdfor else http://anciv.org/PS_cache/physics/pdf/0702/0702063v1.pdf fordetails.

This differential refraction is independent of the photon's pathlengththrough the Faraday medium and occurs within a few wavelengths of theboundary. The Verdet constant V represents an optical parameter thatdescribes the strength of Faraday rotation from interaction betweenlight and a magnetic field for a particular material, named for Frenchphysicist Émile Verdet.

FIG. 1 depicts an optical diagram 100 with accompanying coordinatereference frame 110 for a monochromatic linearly polarized coherentlaser beam 120 of a wavelength λ. The beam 120 forms either aclassically intense continuum or a stream of single photons that isincident from a Faraday medium 130 to a secondary medium 140 ofnegligible Verdet constant separated by an interface boundary 150. TheFaraday and secondary media 130 and 140 have respective indices ofrefraction n_(±) and n₀. The subscripts plus (+) and minus (−)respectively correspond to right and left circular polarized radiation.The diagram 100 illustrates the paths taken by the light beam 120 at theinterface between the Faraday medium 130 and the secondary medium 140with negligible Verdet constant.

The beam 120 has an angle of incidence of θ₀ from the normal to theinterface 150. In the presence of a longitudinal magnetic field {rightarrow over (B)} (having strength B), the beam 120 refracts at anglesθ_(±) from normal at the interface 150 into two circularly polarizedbeams having an angular divergence δ approximated as:

$\begin{matrix}{{\delta \approx {{- \frac{\lambda \; \sin \; \theta_{0}}{\pi \; n_{0}\cos \; \theta}}{VB}}},} & (1)\end{matrix}$

where angle

$\theta = {\frac{1}{2}\left( {\theta_{+} + \theta_{-}} \right)}$

is the average of θ₊ and θ⁻, which are respectively the right- andleft-circularized refraction angles, and V is the Verdet constant forthe Faraday medium 130.

As shown, the incident beam's direction of propagation determines they-axis of the reference frame 110. The x-axis is in the plane containingthe beam 120 and the normal to the interface at the point of incidence.The origin of the reference frame 110 is defined by the perpendicularintersection of the x-axis with the y-axis at the interface 150. Theusual z-axis (into the plane) with positive direction {circumflex over(z)}={circumflex over (x)}×ŷ completes the reference frame 110.

The longitudinal magnetic field B is assumed to be present and parallelto the positive y-axis. If the photon distribution of the incident beam120 has a Gaussian distribution symmetric about the positive y-axis withmean value at x=0, then the refracted beams exhibit Gaussiandistributions that are symmetric about their refracted paths which arealong the vectors {right arrow over (u)}_(±) in the x−y plane of thereference frame 110. More specifically, the refracted beams exhibitphoton distribution mean values which are rotationally displaced aroundthe z-axis through distinct angles θ_(±)−θ₀ from the positive y-axis inthe direction of vectors {right arrow over (u)}_(±) in the x-y plane,respectively.

This refraction process can be described from a quantum mechanicalmeasurement perspective using the mean value of the intensitydistribution profile produced by a detector as a measurement pointer.This description maintains validity for both a single photon stream anda classically intense beam.

In particular, an Hermitean operator Â can be constructed and used toform a Hamiltonian operator Ĥ that describes a photon-interfaceinteraction which produces the required geometry of the refractionprocess. Let |+

and |−

be the right and left circular polarization eigenstates, respectively,of the photon circular polarization operator {circumflex over (σ)} whichobey the eigenvalue equation:

{circumflex over (σ)}|±

=±|±

  (2)

and have the orthogonality properties:

±|±

=1

and

±|∓

=0.  (3)

One can define the “which path” operator Â as:

Â≡(θ ₊−θ₀)|+

+|+(θ⁻−θ₀)|−

−|  (4)

and the associated interaction Hamiltonian H can be expressed as:

Ĥ=ÂĴ _(z)δ(t−t ₀).  (5)

Here the Dirac delta function δ(t−t₀) encodes the fact that therefraction occurs within a few wavelengths of the interface 150 bymodeling the refraction effectively as an impulsive interaction betweena photon of the beam 120 and the interface 150 at time t₀. The “whichpath” operator Â accounts for the refractive angular displacements ofthe initial photon beam 120 at the interface 150. The operator Ĵ_(z)constitutes the measurement pointer's z-component of angular momentum,and couples the refractive angular displacements to the measurementpointer. One can note that:

└Â,Ĵ _(z)┘=0,  (6)

and that |±

are eigenstates of Â with respective eigenvalues (θ_(±)−θ₀).

FIG. 2 shows an elevation schematic 200 of an apparatus that employsthis angular divergence. A laser 210 emits a photon beam 220 asanalogous to the beam 120. The beam 220 passes through a pre-selectionpolarizer or polarizer 230 to reach a Faraday medium 240 (in the form ofa prism), analogous to the medium 130. The refracted beam passes apost-selection polarizer or polarizer 250 to reach a detector 260 thatmeasures the intensity distribution of the refracted polarizationpost-selected light beams.

For an initial photon polarization state |ψ_(i)

, i.e., the pre-selected state, and an initial (Gaussian) pointer state|φ

, the initial state of the combined pre-selected system and measurementpointer prior to the interaction at the interface 150 at time t₀constitutes the tensor product state |ψ_(i)

|φ

. Note that the beam 220 has passed through the pre-selection filter 230prior to its entry into the Faraday medium 240.

Immediately following the measurement's impulsive interaction, thecombined system is in the state:

$\begin{matrix}{\begin{matrix}{{\Psi\rangle} = {^{{- \frac{i}{\hslash}}{\int{\hat{H}{t}}}}{\psi_{i}\rangle}{\phi\rangle}}} \\{= {^{{- \frac{i}{\hslash}}\hat{A}{\hat{J}}_{z}}{\psi_{i}\rangle}{\phi\rangle}}}\end{matrix},} & (7)\end{matrix}$

where use has been made of the fact that the integral of the deltafunction is:

∫δ(t−t ₀)dt=1.  (8)

Now let the initial polarization state be expressed as:

|ψ_(i)

=a|+

+b|−

,  (9)

in which a and b are complex numbers that satisfy the condition

ψ_(i)|ψ_(i)

=1, and rewrite eqn. (7) as:

$\begin{matrix}{{\Psi\rangle} = {{^{{- \frac{i}{\hslash}}\hat{A}{\hat{J}}_{z}}\left( {{a{ + \rangle}} + {b{ - \rangle}}} \right)}{{\phi\rangle}.}}} & (10)\end{matrix}$

Because of the orthogonality proportion of eqn. (9), the n^(th) power ofÂ assumes the form:

Â ^(n)=(θ₊−θ₀)^(n)|+

+|+(θ⁻−θ₀)^(n)|−

−|,n=0,1,2, . . .   (11)

Then the exponential term of the system state of eqn. (7) can be writtenas:

$\begin{matrix}\begin{matrix}{^{{- \frac{i}{\hslash}}\hat{A}\; {\hat{J}}_{z}} = {\sum\limits_{n = 0}^{\infty}\frac{\left\lbrack {{- \frac{i}{\hslash}}\hat{A}\; {\hat{J}}_{z}} \right\rbrack^{n}}{n!}}} \\{= {{\sum\limits_{n = 0}^{\infty}{\frac{\left\lbrack {{- \frac{i}{\hslash}}\left( {\theta_{+} - \theta_{-}} \right)\; {\hat{J}}_{z}} \right\rbrack^{n}}{n!}{ + \rangle}{\langle + }}} +}} \\{{\sum\limits_{n = 0}^{\infty}{\frac{\left\lbrack {{- \frac{i}{\hslash}}\left( {\theta_{-} - \theta_{0}} \right)\; {\hat{J}}_{z}} \right\rbrack^{n}}{n!}{ - \rangle}{\langle - }}}} \\{{= {{^{{- \frac{i}{\hslash}}{({\theta_{+} - \theta_{0}})}{\hat{J}}_{z}}{ + \rangle}{\langle + }} + {^{{- \frac{i}{\hslash}}{({\theta_{-} - \theta_{0}})}{\hat{J}}_{z}}{ - \rangle}{\langle - }}}},}\end{matrix} & (12)\end{matrix}$

where i=√{square root over (−1)} is the imaginary unit and

$\hslash = \frac{h}{2\pi}$

represents the reduced Planck constant. This result correlatesrefraction angle rotations with polarization.

The exponential operators constitute the rotation operators {circumflexover (R)}_(z):

$\begin{matrix}{^{{- \frac{i}{\hslash}}{({\theta_{\pm} - \theta_{0}})}{\hat{J}}_{z}} = {{{\hat{R}}_{z}\left( {\theta_{\pm} - \theta_{0}} \right)} \equiv {{\hat{R}}_{z}^{\pm}.}}} & (13)\end{matrix}$

These operators rotate the x- and y-axes through angles (θ_(±)−θ₀)around the z-axis of the reference frame 110. The rotation notation isconsistent with the convention used by A. Messiah, Quantum Mechanics, v.2, p. 1068 (1961).

This enables the system state in the {|{right arrow over (r)}

} representation to be rewritten as:

{right arrow over (r)}|Ψ

=a|+

{right arrow over (r)}|{circumflex over (R)} _(z) ⁺ |φ

+b|−

{right arrow over (r)}|{circumflex over (R)} _(z) ⁻|φ

.  (14)

The associated pointer state distribution in the {|{right arrow over(r)}

}-representation is then:

|

{right arrow over (r)}Ψ

| ² =|a| ² |

{right arrow over (r)}|{circumflex over (R)} _(z) ⁺

^(φ|) ² +|b| ² |

{right arrow over (r)}|{circumflex over (R)} _(z) ⁻|

φ|²,  (15)

and clearly corresponds to a sum of two Gaussian distributions |

{right arrow over (r)}|{circumflex over (R)}_(z) ^(±)|φ

|² which are each symmetrically distributed about the vectors {rightarrow over (u)}_(±), respectively.

A final photon polarization state |ψ_(f)

that is post-selected can be expressed as:

|ψ_(f)

=c|+

+d|−

  (16)

in which c and d represent complex numbers that satisfy the condition

ψ_(f)|ψ_(f)

=1. Note that the post-selection polarizer 250 receives the beam afterrefraction by the Faraday medium 240. From this, the resulting pointerstate becomes:

|Φ

≡

ψ_(f)|Ψ

=ac*{circumflex over (R)} _(z) ⁺|φ

+bd*{circumflex over (R)} _(z) ⁻|φ

,  (17)

in which the asterisk denotes the complex conjugate, and its {|{rightarrow over (r)}

}-representation distribution is:

|

{right arrow over (r)}|Φ

| ² =|ac*| ² |

{right arrow over (r)}|{circumflex over (R)} _(z) ⁺|φ

|² +|bd*| ²|

{right arrow over (r)}|{circumflex over (R)} _(z) ⁻φ

|²+2Reac*bd*

{right arrow over (r)}|{circumflex over (R)} _(z) ⁺|φ

{right arrow over (r)}|{circumflex over (R)} ₂ ⁻|φ

*.  (18)

One may observe that although eqn. (18) constitutes a sum of twoGaussian distributions that are symmetrically distributed around vector{right arrow over (u)}_(±), unlike eqn. (15), this distribution alsocontains an interference term. Careful manipulation of this interferenceterm can be described herein that produces the desired amplificationeffect.

In contrast to a strong measurement, a weak measurement of the “whichpath” operator Â occurs when the uncertainty Δθ in the pointer'srotation angle is much greater than the separation between Â'seigenvalues, and when the interaction between a photon and the pointeris sufficiently weak so that the system remains essentially undisturbedby that interaction. In this case, the post-selected pointer state isrepresented as:

$\begin{matrix}{\begin{matrix}{{\Phi\rangle} = {{\langle{\psi_{f}{^{{- \frac{i}{\hslash}}\hat{A}{\hat{J}}_{z}}}\psi_{i}}\rangle}{\phi\rangle}}} \\{\approx {{\langle{\psi_{f}{\left( {1 - {\frac{i}{\hslash}\hat{A}{\hat{J}}_{z}}} \right)}\psi_{i}}\rangle}{\phi\rangle}}} \\{\approx {{\langle{\psi_{f}\psi_{i}}\rangle}^{{- \frac{i}{\hslash}}A_{w}{\hat{J}}_{z}}{\phi\rangle}}}\end{matrix},} & (19)\end{matrix}$

or else as:

{right arrow over (r)}|Φ

≈

ψ_(f)|ψ_(i)

{right arrow over (r)}|{circumflex over (R)} _(z)(ReA _(w))|φ

,  (20)

where the quantity A_(w) is expressed as:

$\begin{matrix}{{A_{w} = \frac{\langle{\psi_{f}{\hat{A}}\psi_{i}}\rangle}{{{\langle\psi_{f}}\psi_{i}}\rangle}},} & (21)\end{matrix}$

and constitutes the weak value of operator Â. Note that rotation angleA_(w) is generally a complex value that can be directly calculated fromthe associated theory. One may also note that in response to |ψ_(i)

and |ψ_(f)

being nearly orthogonal, the real value ReA_(w) can lie far outside thespectrum of eigenvalues for Â.

The pointer state distribution for eqn. (20) is:

|

{right arrow over (r)}|Φ

| ²≈

ψ_(f)|ψ_(i)

|²|

{right arrow over (r)}|{circumflex over (R)} _(z)(ReÂ _(w))|φ

|²,  (22)

and corresponds to a broad distribution that is symmetric around avector in the x-y plane. That vector can be determined by a rotation ofthe x- and y-axes through an angle ReA_(w) about the z-axis. In orderthat eqn. (20) be valid, both of the two following general weaknessconditions for the uncertainty in the pointer rotation angle must besatisfied:

$\begin{matrix}{{{(a)\mspace{14mu} \Delta \; \theta}{{A_{w}}\mspace{14mu} {{and}(b)}\mspace{14mu} \Delta \; \theta}\left\{ {\min\limits_{({{n = 2},3,\ldots}\;)}{\frac{A_{w}}{\left( A^{n} \right)}}^{\frac{1}{n - 1}}} \right\}^{- 1}},} & (23)\end{matrix}$

as reported by I. M. Duck et al., “The sense in which ‘weak measurement’of a spin-½ particle's spin component yields a value 100” in Phys. Rev.D 40, 2112-17 (1989). See http://prd.aps.org/pdf/PRD/v40/i6/p2112_(—)1for details.

Use of the above pre- and post-selected states |ψ_(i)

and |ψ_(f)

—along with eqn. (11)—provides the following scalar expression for theweak value of the n^(th) moment of “which path” operator Â:

$\begin{matrix}{{\left( A^{n} \right)_{w} = \frac{{a\mspace{11mu} {c^{*}\left( {\theta_{+} - \theta_{0}} \right)}^{n}} + {b\mspace{11mu} {d^{*}\left( {\theta_{-} - \theta_{0}} \right)}^{n}}}{{a\mspace{11mu} c^{*}} + {b\mspace{11mu} d^{*}}}},} & (24)\end{matrix}$

where c* and d* represent complex conjugates of c and d.

When n=1, then the first moment corresponds to the pointer's peakintensity. The first moment is:

$\begin{matrix}{A_{w} = {\frac{{a\mspace{11mu} {c^{*}\left( {\theta_{+} - \theta_{0}} \right)}} + {b\mspace{11mu} {d^{*}\left( {\theta_{-} - \theta_{0}} \right)}}}{{a\mspace{11mu} c^{*}} + {b\mspace{11mu} d^{*}}}.}} & (25)\end{matrix}$

When the transmission axis of the pre-selection polarizer 230 is set sothat:

a=sin φ,

and

b=cos φ,  (26)

and that of the post-selection polarizer 250 is set so that:

c=cos χ,

and

d=−sin χ,  (27)

then the quantity A_(w) becomes:

$\begin{matrix}{A_{w} = {{{Re}\mspace{11mu} A_{w}} = {\frac{{\left( {\theta_{+} - \theta_{0}} \right)\sin \; \varphi \; \cos \; \chi} - {\left( {\theta_{-} - \theta_{0}} \right)\cos \; \varphi \; \sin \; \chi}}{\sin \left( {\varphi - \chi} \right)}.}}} & (28)\end{matrix}$

One can observe from this that the absolute value of the “which path”scalar |A_(w)| can be made arbitrarily large by choosing φ≈χ, i.e.,separated by a small difference term ε. In particular, let χ=φ−ε andφ=π/4 (in which case the pre-selected state is linearly polarized in thex-direction). Consequently.

$\begin{matrix}{{{(a)\mspace{14mu} \sin \; \varphi} = {{\cos \; \varphi} = {\sqrt{2}/2}}},{{(b)\mspace{14mu} \cos \; \chi} = {\frac{\sqrt{2}}{2}\left( {{\cos \; ɛ} + {\sin \; ɛ}} \right)}},{{(c)\mspace{14mu} \sin \; \chi} = {\frac{\sqrt{2}}{2}\left( {{\cos \; ɛ} - {\sin \; ɛ}} \right)}},{{{{and}(d)}\mspace{14mu} {\sin \left( {\varphi - \chi} \right)}} = {\sin \; {ɛ.}}}} & (29)\end{matrix}$

The previous relation from eqn. (28) for the amplified pointer rotationangle associated with the post-selected circularly polarized beams thenbecomes:

$\begin{matrix}{{A_{w} = \frac{{\left( {\theta_{+} - \theta_{0}} \right)\left( {{\cos \; ɛ} + {\sin \; ɛ}} \right)} - {\left( {\theta_{-} - \theta_{0}} \right)\left( {{\cos \; ɛ} - {\sin \; ɛ}} \right)}}{2\sin \; ɛ}},} & (30)\end{matrix}$

which can be rewritten (by simplifying grouped terms) alternatively as:

$\begin{matrix}{A_{w} = {\frac{{\left( {\theta_{+} - \theta_{-}} \right)\cos \; ɛ} + {\left\lbrack {\left( {\theta_{+} + \theta_{-}} \right) - {2\theta_{0}}} \right\rbrack \sin \; ɛ}}{2\sin \; ɛ}.}} & (31)\end{matrix}$

This quantity is the pointer rotation angle, which can be convenientlyrelated to the angular divergence δ and the difference term ε:

$\begin{matrix}{A_{w} = {\frac{\delta}{2\tan \; ɛ} + {\left( {\theta - \theta_{0}} \right).}}} & (32)\end{matrix}$

Note that for small difference such that:

0<ε<<1,  (33)

then the rotation angle becomes arbitrarily large in magnitude and canbe approximated as:

$\begin{matrix}{{A_{w} \approx \frac{\delta}{2\; ɛ}},} & (34)\end{matrix}$

and because of this, ε can be called the amplification factor

The weakness condition constraint follows when eqns. (24) and (25) canbe used to obtain the associated weakness condition when incorporating0<ε<<1 from eqn. (33) into inequalities eqn. (23), along with selectionsfor a, b, c, d, φ, χ. These steps yield:

$\begin{matrix}{{(a)\mspace{14mu} {\Delta\theta}}{\frac{\delta }{2\; ɛ}\mspace{14mu} {and}\mspace{14mu} (b)\mspace{14mu} {\Delta\theta}}{2{{{\theta - \theta_{0}}}.}}} & (35)\end{matrix}$

Here use is made of the fact that for 0<ε<<1 being sufficiently small,then:

$\begin{matrix}{\begin{matrix}{{\min\limits_{({{n = 2},3,\ldots})}{\frac{A_{w}}{\left( A^{n} \right)}}^{\frac{1}{n - 1}}} \approx {\min\limits_{({{n = 2},3,\ldots})}{\frac{\left( {\theta_{+} - \theta_{0}} \right) - \left( {\theta_{-} - \theta_{0}} \right)}{\left( {\theta_{+} - \theta_{0}} \right)^{n} - \left( {\theta_{-} - \theta_{0}} \right)^{n}}}^{\frac{1}{n - 1}}}} \\{= {\frac{\left( {\theta_{+} - \theta_{0}} \right) - \left( {\theta_{-} - \theta_{0}} \right)}{\left( {\theta_{+} - \theta_{0}} \right)^{2} - \left( {\theta_{-} - \theta_{0}} \right)^{2}}}} \\{= {{\left( {\theta_{+} - \theta_{0}} \right) - \left( {\theta_{-} - \theta_{0}} \right)}}^{- 1}} \\{= \left( {2{{\theta - \theta_{0}}}} \right)^{- 1}}\end{matrix}.} & (36)\end{matrix}$

Satisfaction of both conditions (a) and (b) of eqn. (35) requires thatwhen 0<ε<<1 from eqn. (33), then the uncertainty Δθ greatly exceeds theabsolute ratio value:

$\begin{matrix}{{\Delta\theta}{{\frac{\delta}{2ɛ}}.}} & (37)\end{matrix}$

This condition can be satisfied by making the initial Gaussian pointerdistribution width sufficiently large.

Thus, as per eqns. (22) and (34), the rotation of the initial photondistribution axis of symmetry provides an amplified measurement of theangular divergence δ via the weak value of the “which path” operator Â.For a known amplification ε and a measured mean value of the intensitydistribution profile produced by the detector 260 corresponding toA_(w), then angular divergence δ can be estimated from eqn. (34) as:

δ≈2εA _(w).  (38)

The real component of the complex operator A_(w)=ReA_(w) corresponds tothe angle between the direction of the resultant photon distributionpeak and the positive y-axis is measured when there is sufficientknowledge of the value of the other parameters (e.g., θ, n₀, V, etc.)appearing on the right hand side of this expression.

Consider the case where a target wavelength λ, can be expressed as thesum of a known wavelength λ′ and a difference wavelength Δλ:

λ=λ′+Δλ.  (39)

and similarly average refraction angle θ can be expressed as the sum ofa known angle θ′ and a corresponding difference Δθ:

θ=θ′+Δθ,  (40)

so that approximation eqn. (1) can be written as:

$\begin{matrix}{\delta \approx {{- \frac{\left( {\lambda^{\prime} + {\Delta\lambda}} \right)\sin \; \theta_{0}}{\pi \; n_{0}{\cos \left( {\theta^{\prime} + {\Delta\theta}} \right)}}}{VB}}} & (41)\end{matrix}$

or rewritten as the approximation:

$\begin{matrix}{{\delta \approx {{{- \frac{\lambda^{\prime}\sin \; \theta_{0}}{\pi \; n_{0}\cos \; \theta^{\prime}}}{VB}} - {\frac{{\Delta\lambda}\; \sin \; \theta_{0}}{\pi \; n_{0}\cos \; \theta^{\prime}}{VB}}} \equiv {\delta^{\prime} + {\Delta\delta}}},} & (42)\end{matrix}$

where, because Δθ is small, use has been made of the approximation

cos(θ′+Δθ)≈ cos θ′.  (43)

Using eqn. (42) in eqn. (34) renders:

$\begin{matrix}{{A_{w} \approx {\frac{\delta^{\prime}}{2ɛ} + \frac{\Delta\delta}{2ɛ}} \equiv {A_{w}^{\prime} + {\Delta \; A_{w}}}},} & (44)\end{matrix}$

where A′_(w) represents rotation angle corresponding to the mean valueof the photon distribution profile associated with the measurement ofthe known wavelength λ′. Note that the rotation angle A_(w)=A′_(w) whenthere is no wavelength difference or Δλ=0=ΔA_(w).

The apparatus represented by the diagram 200 in FIG. 2 can be used todetect λ=λ′+Δλ from eqn. (39) and estimate Δλ when λ′ is known. In orderto accomplish this, the apparatus must first be calibrated so that itspointer value is A′_(w) when the source is monochromatic with a knownwavelength λ′. In particular, the photon distribution peak A′_(w) isdetermined by enabling monochromatic light of wavelength λ′ to traversethe apparatus that comprises a Faraday medium with Verdet constant V.

In this example, the parameters θ₀, n₀, B are fixed and the polarizersare set per above values to provide an amplification factor ε. Whenlight of wavelength λ (bichromatic or monochromatic) traverses thecalibrated apparatus, then the pointer deviates from the calibratedpointer value A′_(w) by the amount

$\frac{\Delta\delta}{2ɛ}.$

As numerical examples, consider an additional two instances in which themedium 140 with negligible Verdet constant is air. For the firstexample, let the Faraday medium 130 be terbium gallium garnet (formulaTb₃Ga₅O₁₂) which has a Verdet constant V=−134 rad·Tesla⁻¹·m⁻¹ at knownwavelength λ′=632.8 nm (red light) so that

${\Delta \; A_{w}} \approx {21.33\left( \frac{{\Delta\lambda} \cdot B}{ɛ} \right){{rad} \cdot {Tesla}^{- 1} \cdot {m^{- 1}.}}}$

If for this first example, the amplification is ε=10⁻⁴, the wavelengthdifference is Δλ=1 pm and magnetic field strength is B=1 Tesla, thenrotation angle difference is ΔA_(w)≈21.33 μrad, provided that theassociated weakness condition of minimum rotation angle difference

${{\Delta\theta}{\frac{1.35}{ɛ} \times 10^{- 5}{rad}}} = {0.135\mspace{11mu} {rad}}$

is satisfied. If the detector 260 is 1 m (one meter) from the Faradaymedium 240, then the pointer is translated by −21 μm in the detectorplane. Thus, the 1 pm spectrum separation has been amplified by a factorof ˜10⁶ at the detector 260.

For the second example, let the Faraday medium be MR3-2 Faraday rotatorglass which has a Verdet constant V=−31.4 rad·Tesla⁻¹·m⁻¹ at knownwavelength λ′=1064 nm (and 20° C. temperature) so that the rotationangle difference is

${\Delta \; A_{w}} \approx {4.997\left( \frac{{\Delta\lambda} \cdot B}{ɛ} \right){{rad} \cdot {Tesla}^{- 1} \cdot {m^{- 1}.}}}$

If ε=10⁻³, Δλ=1 nm and B=1 Tesla, then ΔA_(w)≈4.997 μrad provided thatthe associated weakness condition

${{\Delta\theta}{\frac{5.317}{ɛ} \times 10^{- 6}{rad}}} = {5.317\mspace{11mu} m\; {rad}}$

is satisfied.

If the detector 260 is 1 m (one meter) distant from the Faraday medium,then the pointer translates by ˜5.0 μm in the detector plane. Thus, the1 nm spectral separation has been amplified by a factor of ˜10³ at thedetector 260 for the medium 140 with negligible Verdet constant (andunitary refraction index) being air.

As described above, known beam wavelength λ′, magnetic field strength Band angle-of-incidence θ₀ are established á priori. The refraction angleθ′ represents the average of the refraction angles for the circularlypolarized beams determined for the known wavelength λ′ based on theindices of refraction n_(±) of the Faraday medium 130.

For small differences such that Δλ≡λ−λ′<<λ between unknown and knownwavelengths, the average angular refraction angle difference between theunknown (i.e., target) and known refractions Δθ≡θ−θ′ is small. Thisenables the average refraction angle to be reasonably approximated bythe known value as θ≈θ′. The approximation cos θ≈ cos θ′ can be made,because of the relation:

cos(θ′+Δθ)=cos θ′ cos Δθ−sin θ′ sin Δθ≈ cos θ′−Δθ sin θ′≈ cos θ′.  (45)

The distribution peak rotation angle A′_(w) corresponding to the knownwavelength λ′ can be established from a calibration measurement. For asmall amplification factor E such that 0<ε<<1, the rotation angle A_(w)corresponding to the unknown wavelength λ provides an estimate of thedivergence δ from eqn. (38) as δ=2εA_(w). Similarly, the known rotatorangle A′_(w) the estimate for the corresponding divergence δ′.

In response to a small change in wavelength from the known λ′ to anunknown λvalue, the measured rotator angle becomes A_(w)=A′_(w)+ΔA_(w)with the difference ΔA_(w) corresponding to the change in measuredrotation angle due to the offset wavelength that provides a measure ofchange in dispersion angle from eqn. (42) as Δδ. Subtracting thecalibrated values and rearranging terms enables the wavelengthdifference to be determined as:

$\begin{matrix}{{\Delta\lambda} = {{- \frac{\pi \; n_{0}{\Delta\delta cos}\; \theta^{\prime}}{{VB}\; \sin \; \theta_{0}}} \approx {- {\frac{2{ɛ\pi}\; n_{0}\Delta \; A_{w}\cos \; \theta^{\prime}}{{VB}\; \sin \; \theta_{0}}.}}}} & (46)\end{matrix}$

While certain features of the embodiments of the invention have beenillustrated as described herein, many modifications, substitutions,changes and equivalents will now occur to those skilled in the art. Itis, therefore, to be understood that the appended claims are intended tocover all such modifications and changes as fall within the true spiritof the embodiments.

1. An apparatus for determining a target wavelength λ of a target photonbeam, said apparatus comprising: a photon emitter for projecting amonochromatic light beam at the target wavelength λ substantiallyparallel to a magnetic field having strength B and offset fromestablished wavelength λ′ as λ=λ′+Δλ by a wavelength difference ofΔλ<<λ; a pre-selection polarizer through which said light beam passesfrom said emitter as a pre-selection light beam; a prism composed of aFaraday medium having Verdet value V through which said pre-selectionlight beam passes from said pre-selection polarizer and is incident toan interface surface at incidence angle θ₀ to a normal of said surfaceand exists into a secondary medium as first and second circularlypolarized light beams separated by target separation angle δ and havingaverage target refraction angle θ, said secondary medium having index ofrefraction of n₀; a post-selection polarizer through which saidpolarized light beams pass as post-selection light beams; a detector forreceiving said post-selection light beams and measuring target pointerrotation angle A_(w) based on said target separation angle δ; and ananalyzer for determining the target wavelength λ by calculating offsetpointer rotation angle ΔA_(w)=A_(w)−A′_(w), from calibrated pointerrotation angle A′_(w) based on established separation angle δ′ thatcorresponds to said established wavelength λ′, and by estimating saidwavelength difference based on${{\Delta\lambda} \approx {- \frac{2{ɛ\pi}\; n_{0}\Delta \; A_{w}\cos \; \theta^{\prime}}{{VB}\; \sin \; \theta_{0}}}},$in which ε is an amplification factor, and θ′ is average establishedrefraction angle.
 2. The apparatus according to claim 1, wherein saidpointer rotation angle is$A_{w} = \frac{{\left( {\theta_{+} - \theta_{-}} \right)\cos \; ɛ} + {\left\lbrack {\left( {\theta_{+} + \theta_{-}} \right) - {2\theta_{0}}} \right\rbrack \sin \; ɛ}}{2\sin \; ɛ}$in which θ₊ and θ⁻ are respectively right- and left-polarized refractionangles with said average target refraction angle such that θ=½(θ₊+θ⁻).3. The apparatus according to claim 1, wherein said analyzer estimatesdivergence δ≈2εA_(w) for determining the target wavelength.
 4. A methodfor determining a target wavelength λ of a target photon beam, saidmethod comprising: emitting a monochromatic light beam at the targetwavelength λ, substantially parallel to a magnetic field having strengthB and offset from established wavelength λ′ as λ=λ′+Δλ by wavelengthdifference of Δλ<<λ; pre-selection filtering of said monochromatic lightbeam as a pre-selection light beam; refracting said pre-selection lightbeam through a prism composed of a Faraday medium having Verdet value Vsuch that said light beam is incident to an interface surface atincidence angle θ₀ to a normal of said surface and exits into asecondary medium as first and second circularly polarized light beamsseparated by target separation angle δ and having average targetrefraction angle θ, said secondary medium having index of refraction ofn₀; post-selection filtering of said polarized light beams aspost-selection light beams; measuring target pointer rotation angleA_(w) based on said target separation angle δ of said post-selectionlight beams; and determining the target wavelength λ by calculatingoffset pointer rotation angle Δλ=A_(w)−A′_(w) from calibrated pointerrotation angle A′_(w) based on established separation angle δ′ thatcorresponds to said established wavelength λ′, and by estimating saidwavelength difference based on${{\Delta\lambda} \approx {- \frac{2{ɛ\pi}\; n_{0}\Delta \; A_{w}\cos \; \theta^{\prime}}{{VB}\; \sin \; \theta_{0}}}},$in which ε is an amplification factor, and θ′ is average establishedrefraction angle.
 5. The method according to claim 4, wherein saidpointer rotation angle is$A_{w} = \frac{{\left( {\theta_{+} - \theta_{-}} \right)\cos \; ɛ} + {\left\lbrack {\left( {\theta_{+} + \theta_{-}} \right) - {2\theta_{0}}} \right\rbrack \sin \; ɛ}}{2\sin \; ɛ}$in which θ₊ and θ⁻ are respectively right- and left-polarized refractionangles with said average target refraction angle such that$\theta = {\frac{1}{2}{\left( {\theta_{+} + \theta_{-}} \right).}}$ 6.The method according to claim 4, further including estimating divergenceδ≈2εA_(w) to determine the target wavelength.